Multiplying and dividing linkage



Feh 0 A. SVOBODA 2,49%,312

MULTIPLYING AND DIVIDING LINKAGE Filed April 5; 1946 9 Sheets-Sheet lF56. i k

COMBINATiON OF GRID GENERATOR AND TRANS- FORMER LINKAGES.

Y's-l) Ykl Y FIG. 2 FIG. 3

L-s) i?) i?) i? (2) (-l) (0) ATTORNEY Feb. 21, 1950 A. svoBoDA 2,493,312

MULTIPLYING AND DIVIDING LINKAGE Filed April 5, 1 946 9 Sheets-Sheet 2iNVENTOR GRID STRUCTURE OF ANTONIN SVOBODA F I69 5 BY flaw 4 ATTORNEY IFeb. 21, 1950 A. SVOBODA 2,498,312

MULTIPLYING AND DIVIDING LINKAGE Filed April 5, 1946 9 Sheets-Sheet 3INVENTOR ANTONIN SVOEODA Wa d BY ATTORNEY Feb. 21, 1950 A. SVOBODA2,493,312

MULTIPLYING AND DIVIDING LINKAGE 9 Sheets-Sheet 4 Filed April 5, 1946PEG. 8

iNV ENTOR ANTO NIN SVOBODA ATTORNEY Feb. 21, 1950 A. svoBoDA 2,498,312

MULTIPLYING AND DIVIDING LINKAGE Filed April 5, 1946 9 Sheets-Sheet 5 9lNVENTOR ANTONIN SVOBODA BY My ATTORNEY Feb. 21, 1950 A. SVOBODAMULTIPLYING AND DIVIDING LINKAGE 9 Sheets-Sheet 6 Filed April 5, 1946FIG. IO

INVENTOR. ANTONiN SVOBQDA Jimmy Feb. 21, 1950 A. SVOBODA 2,493,312

MULTIPLYING AND DIVIDING LINKAGE Filed April 5, 1946 9 Sheets-Sheet 7 2f o A'\ 6-m f-m Feb. 21, 1950 A. SVOBODA 8, 1

MULTIPLYING AND DIVIDING LINKAGE Filed April 5, 1946 9 Sheets-Sheet 8INVENTOR. ANTONIN SVOBODA JTTORNL) Feb. 21, 1950 A. SVOBODA MULTIPLYINGAND DIVIDING LINKAGE Filed April 5, 1946 9 Sheets-Sheet 9 INVENTORANTONIN SVOBODA ATTORNEY l atenterl ela. 2i, 1956 UNITED STATES PATENTOFFICE MULTIPLYING AND DIVIDING LINKAGE- Antonin Svoboda, Cambridge,Mass.

Application April 5, 1946, Serial No.,659,713

4 Claims. 1

This invention relates .to a computing device and more particularly to acomputing device for solving a predetermined function of two independentvariables.

Computers heretofore known in the art normally are of bulky constructionand are expensive to manufacture. This invention provides a computingdevice which is both economical to manufacture and which is economicalof space.

For general information purposes in connection with the presentinvention, reference is made to the textbook, Computing Mechanisms andLinkages, vol. 27, by Antonin Svoboda, Massachusetts Institute ofTechnology, Radiation Laboratory Series, First Edition 1948, McGraw-HillBook Company, Inc.

An object of this invention is to provide a novel computing device forsolving a' predetermined function of two independent variables.

Another object of this invention is to provide a novel bar linkagemechanism for solving a predetermined function of two independentvariables,

A further object of this invention is to provide a novel linkagemechanism for solving a predetermined function of two independentvariables consisting of three links pivotally connected at a commonpoint, together with constraining means operatively connected to thefree ends of said links for guiding said free ends adjacent suitablescales, the relative lengths of said links and orientation of saidconstraining means being such that movement of one of said links isproportional to said predetermined function in response to movement ofeither of the other of said links.

These and other objects of this invention will be apparent from thefollowing specification and claims in connection with the accompanying,

drawings in which:

Fig. l is a schematic diagram of a grid generator and transformerlinkage as contemplated by this invention.

Fig. 6 illustrates a mechanization of the grid structure of Fig. 5.

Fig. 7 is a topological transformation of the grid structure of Fig. 5.

Fig. 8 is a mechanization of the grid structure of Fig. 7. I

Fig. 9 illustrates a modification of Fig. 8 having transformer means forlinearizing the readings of the scales of Fig. 8.

Fig. 10 is the grid structure of the relation developed about the center.E2=.'1J3'=1.

Fig. 11 is a grid structure combining the grid structure of Fig. 10 intofour quadrants.

. Fig. 12 is a topological transformation of the grid structure of Fig.11.

Fig. 13 is a mechanization of the grid structure of Fig. 12.

Fig. 14 is a modified mechanization of the grid structure of Fig. 12.

A computer constructed according to the principles of this inventionconsists of a mechanism with two degrees of freedom and at least oneoutput parameter, Xk, functionally related to two input parameters, Xiand X thus To such mechanism may be added functional scales thatestablish relations between. the parameters X1, X Xk and correspondingvariables m1, :m, an, respectively. The mechanism will then serve toestablish a functional relation :Ck f ($1,561) (2) between thesevariables. The device, including mechanism and scales, then mechanizesEquation 2. If this relation of the variables is to be singlevalued, itis necessary that to definite values of the input variables therecorrespond definite values of the input parameters, and that to adefinite value of the output parameter there correspond a definite valueof the output variable.

The scales must then establish relations of the form,

Xj=(XJlJZJ).1Lj 13k: (xxIXk) .Xk where all three operators (but notnecessarily their inverse operators) are single valued. If Equations 3are of linear form the device provides a linear mechanization ofEquation 2, i. e., a mechanization wherein all scales are linear innature.

Bar linkages which mechanize functions of two independent variables ascontemplated b this invention have the advantage of being fiat andsmall, of giving smooth frictionless performance allowing appreciablefeedback, and of being relatively inexpensive to manufacture;

The term-topological transformation asused herein refers to the propertyof a complex variable whereby a system of curves may be trans-- formedfrom one plane into another, while .rer taining certain properties. Thisproperty is similar in nature to the property known as conformalmapping, although topological transformation. is somewhat more generalin scope. Discussion of conformal mapping may be found in textbooks ofhigher mathematics' and the technique has been conveniently-applied inhydrodynamicand aerodynamic problems. Three references 8X?plainingthe.nature-. of conformal mapping are submitted: Sec. 142, page465v of Higher, Mathmaticsfor Engineers and Physicists, I. S. and E. S.Sokolnikoff, McGraw-I-Iill, second edition, 1941; chapter 10, page 89,of;Aerodynamic Theory, volume 1, William E. Durand and, Chapter IV; I-Iydrodynamics, 6th Edition, Sir Horace Lamb Cambridge University Press(London), 1932. The mathematical nomenclature used in the followingdiscussion is substantially the same as that employed in the discussionof conformal mapping or conformal" transformation in' the.aforementioned references;

Simple bar linkages can generate only a rather restricted class offunctions so. that in. order to mechanize a given functional relation itusually becomes necessary to use one' or more simple linkages of twodegrees of freedom combinediwith linkages of one degreeof freedom. Aschematic diagram of a grid generator and transformer link age system isshown in Fig. 1'.

Referring to Fig.- 1, let'G- denote -a simple :bar linkage'with twodegreesof freedom, generatinga function of two independent parameters ofa predetermined" class; By combining such a linkage withtthree linkageshaving: one; degree of freedom,.as shown schematically, in Fig, lit ispossiblexto generate relations of .awideclassbetween parameters X1, X5,Xk.

By considering.structures:v consisting of allnkage with two degrees offreedom, which estabe li'shes a relation (Equation5) between internalThe linkage G, with two degrees.v of freedom shall be referred to as.the gridgeneratorf The linkages T1, T Tr shallbe. referred to astransformers, since. they transform the internal parameters Y,intoexternal. parameters X.

The division of a mechanism into a grid generator and transformers isarbitrary; the breakdown of a given functional relation (Equation 1)into a grid generator relation (Equation 5) and transformer relations(Equation 6) is also arbitrary. The term grid generator for a givenfunction shall therefore be used to denote any linkage with two degreesof freedom which will serve as the linkage G in a mechanization of agiven function. By degrees of freedom is meant the minimum number vofindependent variables necessary to specify a given mechanicalconfiguration.

Transformer linkages increase the field of linearly mechanizablefunctions, but not the field of functions mechanizable in the moregeneral sense. A relation mk=f(mi, $7) mechanized by a grid generator(Equation5 transformer linkages (Equationfi) and functional scales(Equation 3) can be mechanized also by associating the same gridgenerator directly with scales which establish relations.

Transformer linkages ina design thus serve only to change the form .ofthefunctional scalesusually to make them linear. A verysimple gridgenerator may be. used if the transformersare sufiiciently complex,whileanother choice of grid generators may. make-unnecessary the use ;ofone or; more transformers A formal characterization of allfunctionalrela tions'which can be mechanized by use; of a given gridgenerator will nowbepresented. Combining Equations 5 and 7, it isapparent that these relations maybe expressed-as $Ic=f($i,$7')=k(G[i($i) ,1(a3 1) (8) where G is the given gridgeneratorfunctionandc1, 4:1, k are arbitrarysinglevaluedfunctions of their arguments.Conversely, to mechanize a given functional relation $k=f(-Ti, my) (2) agrid generator can be employedmechanizing any function of the formYi=G(Yi, Y =k" {f[i (Yi) (Y,-) (9) where gir j'. ,.k are the inverse ofthe arbitrarysinglervalued functions 51,, 4 1, or, which characterizethe transformer linkages. of the mechanism.

These. relations expressed in Equations Band .9 may also be expressedin-terms of contourlinesof thefunctions f and G. Letcontours ofconstants Yk=.G.(Yi, Y belaid outinthe, (YiY planeand labelled withthecorresponding values, of Ye. as shown in Fig. 2. Now. a change may beintroduced. inv the independent variables definedbythe equationsY1==i(a3i) Yi=ai $1l where 411 and 423 are single-valued functions. ofthese arguments; Replotting: thev contours of, G in the (as, 1131)plane; Fig. 3;, lines of; constant f(.'l3l,$j), as defined by Equation 8are obtained.

' If those-contoursare relabeledwith values, of an given. by

$1W=k [Ya (1].) they' will represent the functional relationZElc=f(-'l3i, my). (12) defined by Equation 8, for a particular choiceof the functions qbi, m, 1 It is thus clear that a given grid generatorcanbe used in mechanizing a. given function of the contours of constantG(Y1, Y can be transformed into those of constants Man, at!) orconversely, by any topographical transformation of the form of Equation10, with relabeling of the contours according to Equation 11.

' While formal relations such as Equations 8 and 9 have beendemonstrated, the graphic presentation of these relations throughcontour lines is of more interest. What is particularly of interest isa. means of characterizing given functions, on the one hand, andavailable grid generators, on the other hand, which will make it clearat once whether or not a given grid generator can be used in mechanizinga given function. In this connection, the idea of grid structure of afunction is of fundamental importance.

The representation of a function of two independent variables by a gridstructure is an extension of the representation by a set of contours ofconstant value of the dependent variable. In order to construct a gridstructure, for example of a functional relation itk=f(w m (13) definedthrough a range of operation (domain D) in the (an, my) plane, see Fig.4 which is a diagram of an ideal grid structure of the function Let S bea point in the domain D, associated with values of the variables whichare denoted by an an .Tk this is to serve as the center of the gridstructure. Through S construct the contour B of constant :tk,

See Fig. 4. Next choose an adjacent contour C. defined by xk=mk (15)This, together with point S, will fix the grid structure which is to beconstructed.

Through S construct the vertical line :m=a: intersecting the contour Cat the point (a n a Through this latter point construct the horizontalline $1=$7' intersecting the contour B at the point (331 ZIIJ( 1), :rkThrough this point, in turn, construct the vertical line (Di=$iintersecting the contour C at the point 501 sap- :cr Continuing thus toextend the steplike The rectangular grid of lines will cover part, butnot always all, of the domain D.

This rectangular grid can now be used to define a system of contours12k=$k (18) which, together with the grid itself, will make up therectangular grid structure of the function, defined with respect to acenter S and a contour C.

, The rectangular grid has been so constructed,

and its lines so numbe ed. that a single contour ZRZI) (19a) passesthrough all grid intersections for which r+s=0 (19b) and a. singlecontour xx=wk (20a) passes through all grid intersections for whichr+s=-1 (20b) There is a class of functions such that, no matter how thecenter S and the contour C are chosen, there will be a single contourmn=xk (18) passing through all grid intersections for which t being anyinteger, positive or negative. Such a. function is said to have idealgrid structure."

An ideal grid structure (defined with respect to a center S and acontour C) will consist of the rectangular grid specified above, plusall the contours ark which pass through the intersections of the grid.Such a grid structure appears as shown in Fig. 4.

It is also possible to describe this grid structure as consisting ofthree families of curves, given by Equations 16, 1'7, and 18, such thatthrough every point of intersection there passes a curve of each family.

As heretofore shown, the topological transformation Yl=i ($1) (7) 1=102/) carries contours of the function Yr=G(1/:, 1/1) (3) in the (Y1, Yplane into contours of the function in the (an, an) plane. It is obviousthat it will carry vertical straight lines in the (Y1, Y plane intovertical straight lines in the (:ca LLj) plane, and horizontal straightlines into horizontal straight lines. Indeed, the idea of a gridstructure has been so defined that if this transformation carries acenter SY in the (Y1 Y plane into a center S1: in the (an, an) plane,and a contour CY into a contour Cx, then it carries the complete gridstructure of the function G(Yi, Y defined with respect to Sr and Cy,into the grid structure of the function firm, an) defined with respectto S1: and Cx. The values of the variables associated with the gridlines and contours will be transformed according to Equations 10 and 11,but the indices r, s, t, will be unchanged. 4

It is apparent from the foregoing that a given grid generator can beused in the exact mechanization of a given function if, and only if,there exists a topological transformation, of the form of Equation 7,which carries each grid structure of the function G(Yi Y into acorresponding grid structure of the given function )(an In practice,however, it is only necessary to show that some grid structure of thefunction G'(Yi Y with sufficiently small meshes, can be thus transformedinto a corresponding grid structure of the function f(:n, 2: with errorswithin specified tolerances.

From the foregoing discussion, it is apparent that the grid structure ofa functional relation has been defined as a system of lines in the (m,an) plane: straight lines representing con-- stant,lvaluesrof; anaemiwhich: unit as .rectane gulargrid, and a superimposed family of contoursof constant ark.

Byusing grid structures and'topologicaltransformations as explained.which serve as an intersection nomogram representinga given function,such 'a function cam be mechanized. That is to.=say, such topologicaltransformations of grid structures are employed for mechanizingfunctions of two independent variables; I

Consider; for example, the grid structure of the relation 171::1115/1:1; (22) assh'own inFlg. 5; The'spacingsofthe rect'amgular grid lineschangeiincgeometrical sequence; the fixed ratiobeing 1.25. The contours.oflconstant :ck areradial lines; correspondingivalue of'ak for eachlinejtisthevalue Of...'L'j atiit'sintersection" with. the'verticalline;wr=11 At" each. point" of this figure it is'possibleto read 'ofi' corresponding valuesof'xr, an, and 331i satisfying" Equation22: Tomechanize this function consider three links pivotally connected at oneend thereof at a common point brmeansof'a pin; Thethree free ends areeach constrained by any-suitable constraining means to move adjacentthreedifi'erent scales in response to"'m'ove-' ment of the'common' pointof connection of the three links. If these linkages and. scalesare soarranged that-one can read'* on" the first scale the value of $1 at theposition of the pin, on the second scale the valueqof x and on the thirdscale the value of .rr,,,then:the device provides a mechanization of thegiven function. In .the present case, the firstscale' should" show thehorizontal displacement; of the pin from the origin, thesecond. scaleitsvertical displace ment; the reading of" the third scale should .lie'proportional to the vertical displacement of the intersection of theradialiline through the pin with a vertical. line.v The. divider. (ormulti-- plier) of Fig.1 6. accomplishes this, .itbeinge the naturalmechanizationofthe grid structureof Fig, .5 The..-actiorr;of. thelinkage of .Eig; 6 is, basedon the proportionality-of .thBJSidESsOf twosimilar triangles-i. Thesev are. trianglesv with; horizontal bases,andvertices .at the centralv pin; the first has :a .base of 1 length- Rand. altitude: Xx,,thesecondabase of length .Xi .and altitude,

thus establishing 'amultiplicatio'n function. where R is a constant-thatdepends on thetype' of the multiplier and its dimensions. The-slidemulti' plier of Fig; 6 is'a schematicmather" than'la" practical design;however; since the, lengths f 5 thesliding surfaces as shown are notgreat enough to prevent self-locking in" allpossible' positions of themechanism; I

Atopological transformation ofthis gridstructure will carry it into aform.(Fig; 7) suggesting a very different type of mechanization; The;horizontal 'linesof Fig; 5 are transformed into? a-fa'mily of 'cir'cles,all-of the sameradiusLi' with centers lying on a straight line C1; The"vertical lines-ofFig. 5=are transformed into "a second --familyofcircles; all of "the same "radius; Lz,-. with centers lying =,.onsthecurved line C2. Finally, the radial lines of Fig. 5 are transformed intoan:Jthird"'Tfamilyvofcurves. Tli'ese are'very s nearly although .notvexactly, circleszwithth' same:- radius, L3; since. theapproximatevcircles intersect at a common point their centers mustlieonanother circle with radius Le -curve Ca in the figure. Ignoring thesmall deviation from circular form of the curves of the thirdfamilmthemech-ianization shown. in Fig. 8 results. This isan" approximate, butquite accurate multiplier on divider. The joint P can be made tolie ona. circle of the first family by placing it-atj one end of a bar PA1with length L1, and fixingthe joint A1 in the center of this circle, online C1.- Conversely, if the joint A1 is constrained to'lie on the lineC1, as by being pivoted to'a slideit will necessarily be always at thecenter: of them;- circle on which P lies. A scale placed along C1 can:thus. be calibrated to: give. the value of a3;- at the position of pinP. In the. same way,'the value of 1:1 can be read 011-9 scale lyingalong the curved. line Cz using as: index the point A2 connectedito" thepinzby a; barof length L2. Finally, values of the quotient :cii can beread on the circular scale C's. Instead of pivoting the bar PAs, oflength L3, toa circular slide, it is possible to constrain the point A3to lie on the curve C3; by a second bar 0A3, also of length L3, pivotedat the-center of this circle. As; shown in the figure,- the index point.has been transferred to'this second bar in an obvious way. 1 its-shownthecomputer-for solving the-function Ik- IIH/Iii disclosed in Fig. 8comprises a star linkage consisting of three links 20, 2|, and 22*havingone end of eachpivotally connected at asco-mmon point P; constrainingmeans-a3 heretofore explained are provided attheiree, ends of links 2%,2i", and122: to cause said ends to follow scales calibrated incorrespondence with the grid structure of Fig. 7 to indicateivalues of33k, in, and mi corresponding to theiuncti'on 'i/ To avoid the use of' acurved slideyandth nonlinearity of the set and'sck scales, a moresatisfactory way for guiding point Azalongv the curve C2 may be devisedand the 33x scale and mi. scale.

readingsmay be linearizedbytransformer lin' ages,- such as harmonictransformersor .lthre bar linkages. V Thislmay be accomplishedasshOWn-in Fig. 9 'iv hich I shows the first linkage multiplier sodesigned as to lie operable through asdomfai'ri including positive andnegative. values of all variables. The point A2 is constrained to followthe curve C2 of Fig; 8' by placingit' on an extension of the central barof a three-ibar linkage a ocfi'yfi. Motion of A2 along C2 produces-amonresponding rotation'of thebar a Bofthe threebarlinkage. A harmonictransformer converts this rotation into linearized readings on the $1scale. To linearize the ra scale, the rotation oftlie bar 0A3 isconverted into linear motion of a slide by means of aharmonictransformer. .A complete discussion of the apparatus ofv :9 will. bevfound in ,co-pending' application No. 658,598, filed April 1, 1946; I gBy using the system outlined in the foregoing discussion, a star linkagecan be used in designing, a multiplier mechanizing'the relation .1; $11172 I I In.Fig;.. m the grid structure" for this-relation ,is developedabout the center xz=zcs=l, with the line 1:1:125 chosen'as the con-tourC; The step structure between the contours B and C approaches the originin an infinite number of steps. The grid is ideal, and includes thecontours x1=(1.25)', with 1' taking on all integral values; as rthesecontours approach the horizontal axis, and as r they approach thevertical axis.

To obtain grid structures for this relation in all four quadrants (i.e., to obtain negative values of an, :m and an) it is necessary to use aseparate center for the grid structure in each quadrant. If these fourpoints are chosen in similar positions in the four quadrants,symmetrical with respect to the two axes, and if corresponding contoursC are used, then the four grid structures will approach the coordinateaxes symmetrically. They will then appear to flow smoothly into eachother in crossing these axes, and the whole figure will take on theappearance of the single grid structure shown in Fig. 11. It isimportant to remember that this result is obtained artificially, andthat the coordinate axes are lines of condensation in the gridstructure.

To construct a mechanism from this grid structure, it is necessary totransform the grid structure of Fig. 11 (Cartesian coordinates are, .rx)into an ideal grid structure in which each of the three families ofstraight lines in the original structure is represented by a family ofcongruent circles (Cartesian coordinates ya, ya).

Consider first the family of lines of constant on. In the original gridstructure, all these lines intersect at a common point 0. Such aproperty will not be changed by a topological transformation; in thetransformed grid structure, the corresponding family of circles ofradius L must all intersect at a common point (see Fig. 12). It followsthat the centers of these circles must all lie on a circular arc ab withradius L1 and center at O'. The radius L1 can be chosen at will, as theproblem is independent of the scale of construction; it is usuallyconvenient to take L1 as the unit of length.

The relation between a given straight line of the original gridstructure and the circle into which it is transformed may be establishedby examining the topological transformation in the neighborhood of theorigin 0. In its general form this transformation is (112. 113) If =a112,113)

In the neighborhood of 0, where all variables can be treated as smallquantities, this reduces to on neglect of small quantities of the secondorder. If it is assumed that the transformed grid structure issymmetrical with respect to the horizontal axis, in the neighborhood ofthe origin, then 02a=ca2=0 (25) and 9.9. 1 2 zzy:

11 change in a geometric progression, the slopes at O' of thecorresponding circles must alsov change in geometric progression, and bythe same ratio (1.25). This is true also of the slopes of the radii fromthe origin 0' to the centers of these circles which are the negativereciprocals of the slopes of the circles themselves. By choosing onearbitrary value of 033/022, there can be constructed in the transformedgrid structure, circles corresponding to each of the lines of constant$1 in the original grid structure. In Fig. 12 there are indicated fourof these circles (An, A 1, A 2, A-a) with centers above the horizontalaxis do at points an, a-r, 0-2, a-a; the distinguishing subscripts beingthe r values of the original grid lines, which lie in the second andfourth quadrants of Fig. 11. The corresponding lines in the first andthird quadrants transform into the circles Bo, B-1,.B 2, 3-2, withcenters b0, b 1, b-a, b-s, which are the mirror images of an, 0-1, a z,ass, in the horizontal axis do. The sequence of points at, a 1, a 2,which lies in the domain of positive in has a point of condensation canon the horizontal axis; an extension of the r1scale into the domain ofnegative :81 is provided by the symmetrically placed sequence b0, b 1which has the same point of condensation. This point is the center ofthe circle into which the vertical axis of the original grid structureis transformed.

Consideration of .this family of circles will make it clear that thereexists no topological infinity, but the. circles into which it is hereinattempted to transform them may intersect anywhere in the (g2,y3)-plane, if arbitrarily large values of 1' or :01 are admitted. Atbest, it is only possible to establish a topological transformation thatcarries a portion of the original structure into a grid structureconsisting of arcs of circles.

Consideration is next required of the family of circles of constant as.Since the original lines x3=c were symmetrical with respect to thehorizontal axis, it is natural to give the transformedcircles similarsymmetry; their centers must lie on the horizontal axis ed. The verticalaxis 33:0 is already known to be transformed into a circle A-oo ofradius L1 with center at a-m. It follows that this second family ofcircles must have the same radius. as the first: L3=L1. Since the linesx3==x3 converge on r3=0 as tthe point a c must be a point ofcondensation on the an scale as wellas on the m1 scale. The lines of theoriginal grid intersect the an axis at an the transformed circles mustintersect the y:- axis at points determined by $3 =\//(O,y3 (27) or, inthe immediate neighborhood of 0', by

r3 =c33ys (28) The values of 213" go to zero in a geometricalprogression (ratio 1.25) as t- It follows that the sequence of values1113" approaches zero,

sion is exact, rather than an approximation valid only in theneighborhood of O; that is, assume that Equation 28 is valid for allvalues of t. Then, after an'arbitrary value of 033 is chosen,

a circle corresponding to any 33s-1in8 of the o and C33) determinetwo-families'of intersecting circles, and therebydetermine'completelythe natureof the topological transformation. It is immediately evidentthatthis type of'transformation does not'ha-ve the desirous ,characterof providing'that the v:rz-con-tours"areualso transformed into a familyof circles witha common radius'Lz. ,In'i'Fig. 12 there appear sixty-fourpoints of intersection ofthe-m1-pandas-contours,

distinguished by small circles. These are the transformed positions ofthe intersection in the original grid, and through them. must pass'thetransformed contours ,of "constant .mz. It will be observed thattheintersections in the lower.

half of the grid lie on curves that-are concave upward, whereas thosei-n'the upper half'liein curves .(not shown) which are concave'downward.By'the symmetry of construction, the straight line 502:0, .must'betransformed, not'into acircle, but into the straight line y=; the radiiofcurvature of the other-zcz-contours increase as they approachthis-limiting straight, line. The transformed idea1 grid structureis..not"that'of a star linkage, Thisgrid structure, however, can beapproximated .by the nonideal grid structure of a star linkage.replacing the system of 122-0011 tours, of the'ideal; grid structure bya system of approximating. circles of the same radius.

It is possible to pass circles verynearly'through the intersections inthe lower half plane of Fig. 12 by .choosing a mean value L2 for theradius of curvatureand locating the-centers of the circles in the upperhalf plane. This willestablish the general positionbf the :m-scaleandwill make it necessary to 'pass' through the-intersections of theupperhalf plane circularyarcs that are concave upwardathe'fitcthere'cannot bevery exact, and the errors of constructions must be split as well aspossible.

' The best way to ldothis is to construct acircle through one set ofintersections toestablishthe radiusln. (In Fig/12; E0 is the circleinquesticn,v and the radius chosen is justequal to L1 and L3.) With.thisra'dius, .arcs can be constructed about each of the known gridintersections. If the grid structure underconstruction were to be ideal,the arcs characterized by a given value of s.=.t-.r would all intersect.in, a common point. In the present case, instead of points ofintersection, there exist more or less diffuse points of intersection,within which there can be located with some. degree of arbitrariness.the centers of gridrcircles. Thisarbitrariness can be :used togood-advantage. 'If asimple mechanization-ofathe-grid structure is to bepossible, the xzescale must lie. inaa simple curve, preferably astraight line-or.- a circle. .In the present case the regions of;intersection lie roughly on a circle. I-n'particulanthe circular are c,with center at Q, passes nicely through all regions ofintersecticnexcept-iorrthree at theextreme ends. This circle will betakenuas-thescz-scale on vthisscale, and as near to the centers of'theregions of inter- 61, e0, e 1 of the grid structure circles E1,E11,"E-1, inthe lower half plane, and the centers f1, in, 1 1, structurecircles F1,"Fn, F 1, in the upper half plane. The grid structure circlesmust convergeon a circle E QQ =F m through the origin -O'. The center ofthe circle Eoo 'is Efthe pointof condensation of the point sequence asin the, positive domain of $2, and-of the "point sequencers-in thenegative domain, as'sa-wyit is the zero point "on them scale. Thiscompletes the determination of the constants of 'thestar 1 linkage.

"'Ihe' grid structure of Fig. '12 is mechanized by 'theapproximatemultiplier shown in'Fig. 13. This mechanism consists of a star gridgenera-tor with the arms L1, L2, and L3, meeting at a-comrnonpOint-P.'The'free end of L1 is forced to move along a circle a with the center0', and

the freeend oils-along a circle e with center Qyby-arms R1 andRa'respectively. The free end-of L3 'isconstrained-to move in a straightline bythe slide ed. 'It follows from the foregoing discussion-that thelengths of L1,-L3, and Rr-must be equalyLz is also of the same length,but only accidentally. To use the :apparaltusof Fig. -13 as amultiplier, "the scales must be calibrated in terms of the variables-r1,1112, ms, related to the indices 1", s, and-t. The scale pointsof Fig.12 thus occur forvalues Offlindz, :03, whichchange in geometricprogression; these are the "scale calibrations shown in Fig. 1-3. Itwill be observed that the device computes-the relations a13=m1m2 over aconsiderable range with a'rather high degree of accuracy.

Referring now to-Fig, 14, there is shown a computing device which isbasicallythe same as that shown inFig. 13 except that increasedaccuracy-over the entire range of all scaleshas having-acenter .C,-.by.arms I26. and I25, respectively. (It will be observed that the positionof thecenter O of the 5:2 scale of Fig. 14 is some what difierent fromthat'of the center Q of the 232:"80318 .ofiFig. 13. This changeincreases the The vireeendro'f arm. l20jisiconstrained. to "movemar-straight. line adjacent the "ma-scale l'24 by a slide block .130-

accuracy of ::the :multiplier.)

slidably mounted on base I 21 by two pairs/of rollers i l and 532. Thefree end of arm i2?) is pivotally connected to slide I30 by pin I33. Sim

ilar to the device of Fig. 13, the scales I23, 122

and,l24 are calibratediin terms of the variable 2:1, 1:2 and ms,respectively; namely, in such a manner that the scale divisions changein geometric progression. The ranges of the variablesof the-device ofFig. 13 has been changed'in -ap-- paratus of Fig. 14. The readingson'the'm1-,'w2-,

and'u'cs-scales have changed by the factors afb, and c; respectivelyfthefactorsa, b, and 'c being'controlled by the relation ab=c. .Assuming'R'as a unit of lengthj-the relative dimension of of the grid theelements of the computing device are as follows:

Length of link I I 1.000 R Length of link I20 1.000 R Length of link I001.025 R Length of bar 125 1.310R Length of bar 126 1.000R

line 00 .116 R Angle between the longitudinal axis of scale I24 and line00 13 Radius of scale I22 from point C 1.310 R Radius of scale 123 frompoint 0 1.000R

While particular embodiments of this invention have been disclosed anddescribed, it is to be understood that the scope of the invention is notto be limited except as defined in the appended claims.

What is claimed is:

1. A mechanism for use in a computing mech anism for determining theproduct of the independent variables expressed by the equation m3=x1xacomprising a support, first, second. and third links of substantiallyunit length pivotally connected at a common point, a first scale ofcircular configuration mounted on said support and calibrated toindicate positive and negative values of input variable am, said firstscale having scale divisions which vary in geometric progression oneither side of a zero point, a second scale of circular configurationcalibrated to indicate positive and negative values of input variable:02, said second scale having scale divisions which vary in geometricprogression, a third scale mounted on said support and calibrated toindicate positive and negative values of an, the scale divisions of saidthird scale varying in geometric progression along a straight line, afirst constraining member pivoted to said support for forcing the freeend of said first link to move along said second scale, a secondconstraining member pivoted to said support for forcing the free end ofsaid third link to move along said first scale, and a block slidablymounted for relative parallel motion with respect to said third scale,the free end of said second link being pivotally attached to said blockand forced to move along said third scale thereby, the relative lengthsof said first and second constraining members with respect to said unitlength first, second and third links being 1.1310 and 1.000,respectively.

2. A linkage mechanism for use in computing apparatus, comprising, asupport, first, second and third arms pivotally connected at a commonpoint, first and second circular scales calibrated to indicativepositive and negative values of input variables an and m, respectively,said scales having scale divisions which vary in geometric progressionon either side of zero, and a third scale calibrated to indicatepositive and negative values of a variable quantity :03, said thirdscale having scale divisions which vary in geometric progression along astraight line on either side of zero, a first constraining memberpivoted to said support for forcing the free end of said first arm tomove along said second scale, a second constraining member pivoted tosaid support for forcing the free end of said third arm to move alongsaid first scale, and means for moving the free end of said second armalong said third scale, in which the lengths of the arms andconstraining members measured between pivot points bear the followingrelations to each other when the basis of comparison is taken as unity:

First arm 1.025 Second arm 1.000 Third arm 1.000 First constrainingmember 1.310 Second constraining member 1.000

said dimensions being such that the movement of the constrained end ofsaid second arm along said third scale closely approximates the productof the movement of the constrained ends of said first and third armsalong said second and first scales, respectively, thereby to determinethe relation. m3=anx2, over a substantial range of movement of saidfirst and third arms.

3. A linkage mechanism for use in computing apparatus, comprising aplanar support, first, second and third arms pivotally connectedtogether at a common point, first and second arcuate scales beingpositioned in cooperative relationship with said apparatus, said firstand second scales being calibrated to indicate positive and negativevalues of input variables X1 and X2. respectively, and having scaledimensions which are spaced in accordance with a geometric progressionon either side of zero, said first scale being calibrated from -1 to +1and said second scale being calibrated from 1.25 to +1.25, a third scaleof linear configuration being positioned in cooperative relationshipwith said apparatus and calibrated to indicate positive and negativevalues of an output variable X3, said third scale having scale divisionswhich are spaced in accordance with a geometric progression from 1.25 to+1, a first constraining member pivoted to said support for forcing thefree end of said first arm to move along said second scale, a secondconstraining member pivoted to said support for forcing the free end ofsaid third arm to move along said first scale, and means for moving theunpivoted end of said second arm along said third scale, the lengths ofsaid arms and constraining members measured between pivot points bearingthe following dimensional relation to each other when the basis ofcomparison is taken as unity:

First arm 1.025 Second arm 1.000 Third arm 1.000 First constrainingmember 1.310 Second constraining member 1.000

' said dimensions being such that the movement of the unpivoted end ofsaid second arm closely approximates the product of the movement of theconstrained ends of said first and third arms along said second andfirst scales, respectively, thereby to determine the relation X3=X1X2over the range of movement of said first and third arms, theaforementioned arms, members, and scales having the followingorientation when the basis of comparison is taken as unity:

Distance between the points of attachment to said support of said firstand second constraining members along a line joining said pointsPerpendicular distance between :a -,;line

drawnthrough the points of attachment of said first and secondconstrainingmembersto said support anda line drawnparallel'thereto-through the +1 calibration of tsaidithird scale Anglebetween the said parallel line. drawn through the +1 calibration of saidthird scale and the axis of said third scale--- Distancealong a linedrawn through the points ofv attachment of said first and secondconstraining members tosaid support between the point of attachment-ofsaid second constraining member and the +1 calibration of said thirdscale [1.163 Length of said thirdscale between the +1 and -1 calibrationpointsthereof Angle along said first scale betweenthe +1 and -'1calibration points thereof 44.4 Angie along said second scale betweenthe +1 and '1 calibration points thereof 836 4. A linkage mechanism foruseincomputing apparatus, comprising, a planar support-first, second andthird arms pivotally connected together at a common point, first andsecond arcuate scales, said first and second scales being calibrated toindicate positive and negative values of independent inputvariables X1and X2, respectively, and having scale dimensions thereon which arespaced in accordance with a geometric 'pro- .gression on either side ofa zero position, said first scale being calibrated with valuesfrom -a to+a and said second scale being calibrated with values from ,1.25b to +1.25b, a third scalezof linear configuration being mounted on said supportand having calibrations thereon to indicate positive and negative valuesof'a dependent output variable X3, said third scale havingscale-divisions thereon which are spaced in accordance with a geometricprogression from 1.25c to +0,

1 the relation between the said quantities a, b and c being ab=c, afirst constraining'member pivotally attached at one end to'said supportand at the other end to the free end of saidfirst arm 'to move the freeend of saidfirst arm along said second scale, a second constrainingmember pivotally attached at one end to said support and at the otherend to the free end of said third arm to 'move'the free end of saidthird arm along said first scale, and a sliding member mounted forreciprocating movement in parallel juxtaposition-withsaid third scale,the free end of said second arm being pivotally attached to'said slidingmember, said arms and members having the following relative dimensionsand orientation when the basis of comparison is taken 'as' R,'-R

. being an arbitrary unit of length Length of said first. arm Q1.110251?v Length of said second arm 1.0005.

calibration point of said third scale- .116R Angle between the saidparallel line drawn'through the +0 calibration of said third scale andthe axis of said third scale 113 Distance along a line drawn through thepoints of attachment or said first and second constraining members tosaid support between the point of attachment of said second'constrainingmem- "ber and the +0 calibration point of said third scale 1.1638.Length of said third scale between the +c and -c calibration pointsthereof .446R Angle along said first scale between the +11 and -acalibration points thereof" "443 Angle along said second scale betweenthe +17 and -b calibration points thereofi- 83.6"

thelaforementioned dimensions and orientations providing that themovement of the said free end of said second arm closely approximatesthe prodnet of the movement'of the constrained endsof said first andthird arms along said second and first scales, respectively, thereby todetermine the relation X3=X1X2.

ANTONIN SVOBODA.

REFERENCES CITED The following references are of record in'the file ofthis patent:

UNITED STATES PATENTS Number Name Date 2,229,156 WertheimerJan.'21,.1941 2,272,256 Vogt Feb. 10, 1942 2,394,180 Imm Feb. 5, 1946FOREIGN PATENTS Number Country Date 367,642 Germany Jan. 25,, 1923512,073 Germany -1 Nov. 6,1930 566,565 Germany Dec. 19, 1932 710,028vFrance May 26, 1931 357,940 Italy April 1, 1938 144,893 SwitzerlandMay,,.16,'1931

